IS 10427( Part 2 ): 2006 Indian Standard DESIGN FOR ~DUSTRIAL PART 2 ORTHOGONAL (First EXPERIMENTATION ARRAYS Revision ) ICS 03.120.30 0 BIS 2006 B-U-REAU OF .INDIAN STANDARDS MANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG NEW DELHI 110002 June 2006 Pdce Group 10 Statistical Methods for Quality and Reliability Sectional Committee, MSD 3 FOMWORD This Indian Standard ( Part 2 ) ( First Revision ) was adopted by the Bureau of Indian Standards, afier the drafi finalized by the Statistical Methods for Quality and Reliability Sectional Committee had been approved by the Management and Systems Division Council. Industrial organizations constantly face~he problem of decision making regarding product/process design, process specifications, quality improvement, identification of dominant factors affecting quality, cost reduction, import substitution, etc. In all such problems, one is confronted with several alternatives and one has to choose that alternative which satisfies the requirements at minimum cost. For taking a right decision in all such cases, an experiment may have to be carried out either to discover something about a particular process or to compare the effect of several conditions on the phenomenon under study. The effectiveness of an experiment depends to a large extent on the manner in which the data are collected. The method of data collection may adversely affect the conclusion that can be drawn from the experiment. If, therefore, proper designing of an experiment is not made, no inferences maybe drawn or if drawn may not answer the questions to which the experimenter is seeking an answer. The designing of an.experiment is essenti?.lly the determination of the pattern of observations to be collected. A good experimental design is one that answer efficiently and unambiguously these questions, which are to be resolved and furnishes the required information with a minimum of experimental effort. For this purpose the experiments maybe statistically designed. Part 1 of this standard covers the basic designs, namely, completely randomized design ( CRD ), randomized block designs ( RBD ), latin square designs, balanced incomplete block designs ( BIBD ) and factorial designs. The factorial designs enable evaluation of main effects and interactions and also provide more efficient estimates, Iiowever, one disadvantage with the factorial designs is that it calls for a large number of experiments. It is possible to reduce the numkr of experiments and still estimate most of the important effects. This is achieved by fractional factorial experiments. By carrying out fractional factorial experiments, some information is lost. But when there are several factors, higher order interactions are generally not of much importance and in some cases difficult to interpret. Hence information on these "higher order interactions are deliberately ignored to reduce the number of experiments. Orthogonal array ( OA ) designs, which are discussed in this part, constitute one particulartype of the fractional factorial designs. A special feature of these designs is the associated concept of linear graphing, which enables a scientist or an engineer to design complicated experiments without requiring sophisticated statistical knowledge. The OA designs can meet the needs of various practical situations, such as: a) b) c) studyiwg the effect of various factors having different number of levels, analyzing nested factorial effects when nested factors coexist with some other common factors, and estimating all the main effects along with a few desired lower order interactions. The following changes have been made in this revision: a) b) c) More commonly used designs like L12 (2") for 2" series and L18 (2 x 37) have been included. Table 2 and Table 7 of pre-revised version have been corrected. Other editorial corrections have been incorporated, The other part in the series is: Part 1 Standard designs The composition of the Committee responsible for the formulation of this standai-d is given in Annex G. .1S 10427( Part 2 ): 2006 Indian Standard DESIGN FOR WDUSTNAL PART 2 ORTHOGONAL (First 1 SCOPE EXPERIMENTATION ARRAYS Revision ) 4 USE OF ORTHOGONAL TABLES ARRAY ( OA ) This standard ( Part 2 ) provides methods of planning and conducting experiments using orthogonal array ( OA ) tables when all the factors are either at two levels or at three levels. The procedure is also discussed when some of the factors are at two levels and the remaining at three or four levels. It also describes the procedure for the analysis of the data and selection of the optimum level of each factor. 2 REFERENCES The fotlowing standards contain provisions, which through reference in this text constitute provisions of this standard. At the time of publication, the editions indicated were valid. All standards are subject to revision and parties to agreements based on this standard are encouraged to investigate the possibility of applying the most recent editions of the standards indicated below: IS No. 6200 Title 4.1 In general, an experiment in which all possible Statistical tests of significance: ( Part 1 ): 2003 Part 1 t-, Normal and F-tests 7920 Statistical vocabulary and symbols: ( Part 3 ): 1996 Part 3 Design of experiments 4905:1968 Methods for random sampling combinations of factor levels are realized is calleda full factorial experiment. Therefore, the total number of experiments ( N ) to be conducted is equal to s", wheres is the number of levels of each factor and-n is number of factors, if there are 15 factors, and each factor has 2 levels, then the total number of experiments to be conducted is 2*5. The number of experiments in a factorial experiment is considerably high and sometimes prohibitive in actual use. In fact, orthogonal arrays ( OA ) tables evolved through the concept of fractional replication, that is, sacrificing information about interactions which are usually not very important in an industrial.proj ect, could find itself in sound foating in minimizing the number of experiments. It is seen that while investigating the influence of 15 factors ( each at two levels ), the number of experiments can be reduced to 16 by using OA tables. The effectiveness of using OA tables depends solely Qn the successful selection of the scheme of confounding the interaction effects and on the skilful strategy of choosing the levels of the factors and running the experiment. A prior inform~tion on interactions render a great service to experimenter in this case. 4.2 An orthogonal as L~[ (s)"], where n = number of factors; array is also represented 3 TERMINOLOGY For the purpose of this standard, the definitions given in IS 7920 ( Part 3 ) and the following shall apply. 3.1 Orthogonal Array ( OA ) Tables -- An N x n array ofs symbols is said to be an Orthogonal Array of strength t if every N x t sub-array contains every t-plet ofs- symbols an equal number of times, say k. Thus N=L. St. An OA of strength t is represented as OA ( N, n, S, f ), where N denotes the number of experiments, n denoted the number of factors ands is the number of levels of the factors. It is known that OAS have a close association with fractional factorial experiments. An OA of strength two is an orthogonal main effect plan. 1 s = number of levels of each factor; and N = total number of experiments to be conducted. 4.3 Orthogonal arrays were known as square games in former days. Recently these arrays have been effectively applied in the layout of experiment. An example of orthogonal array of 27 or simply L8( 27) array is given in Table 1, as.per Annex A. 4.3.1 There are 8 experiments in this array. Each column consists of 1 and 2 each four times. When two columns consist of figures 1 and 2 and also they have same number of combinations, that is 1 ),and(2,2) arerepeatedsame (1, 1 ),(1,2),(2, number of times, the two columns are said to be balanced or orthogonal, IS 10427( Part 2 ): 2006 Table 1 Orthogonal Array of27 ( Clause 4.3) 1 ,ment No. (1) 1 2 3 4 5 6 7 8 (2) I 2 1 2 I 2 1 2 2 (3) 1 1 2 2 1 I 2 2 3 (4) 1 2 2 1 1 2 2 1 4 (5) 1 1 1 1 2 2 2 2 5 (6) 1 2 1 2 2 1 2 1 6 (7) 1 ] 2 2 2 2 1 1 7 (8) 1 2 2 1 2 I ] 2 and G2. Against each experimental trial, let the response be recorded as y] , y2 .... yg. In order to compare the effect caused by factor A, the total of responses resulting under conditions A~ and AZ are calculated separately, that is, to sum up the responses in experiment number 1,3, 5, and 7 which were carried out under conditions Al and also sum up the responses in experiment'2, 4, 6, and 8 under conditions A2. Let A, and A2 denote totals of results under conditions A, and A2 respectively. A] `Yl +Y3+Y5+Y7 A2=Y2+Y4+Y6+Y8 Dividing the above results by 4, the average response of A1 and A2 are calculated as: i=(Yl+Y3+Y5+Y7)/4 4,3.2 The necessary condition for an array to be orthogonal is that for all pairs of columns, particular levels appear together an equal number of times. For example, in Table 1, by taking any pair of columns, the level combination ( 1, 1 ); ( 1,2 ); ( 2, 1 ); ( 2,2 ) appear equal number of times, that is, twice. Mathematically, this condition may be written as follows: n,. = `2=(y2+y~+y6+Y*)/4 B, and B2 are similarly compared by the averages responses of the results under condition BI number 1, 2, 5 and 6 ) and B2 ( experiment ( experiment number 3,4,7 and 8 ). Other factors are compared in the same way. `i. x `j N lJ For every combination of ( i, j ) level and every pair of columns where nij = number of times the level combination (i, j ) occurs in any two columns, ni, = number of times the level i occurs in one column, n.j = number of times the levelj occurs in other column, and N= nll nl. total number of experiment in Table 1, -- = N=8 n12 n2. = = n21 n.1 = = n22 n.2 =2 = 4 and The above condition holds good for every combination of levels and every pair of columns. The~efore, Table 1 is an orthogonal array. 4.3.3 One factor can be assigned to a column of Table 1. Let the seven factors assigned to columns 1 to 7 be called A, B, C, D, E, Fand G. For experiment number 1 ( see Table 1 ) all the figures are 1 which means that all the factors in the experiment are in the first level. It is expressed as Al, B,, Cl, D], El, F1 and G,, Lg ( 27) orthogonal arrays is method to carry out 8 experiments to independently compare the effects between A, and AI , B] and B2 ... .. , G1 2 4.3.4 So, even by reducing the size of the experiment it is easy to conclude which factor influences the ultimate response under consideration and should be controlled. The advantage of OA techniques lies in the high reproducibility of the fictorial effect. In OA experiment the difference between the two levels A, and A2 is determined as the average effect while the conditions of.other.factors vary in equal measure. If the influence of A, and A2 to the experimental result is consistent while the conditions of other factors vary, the effect obtained from the experiments using OA tends to be insignificant. On the other hand, if the difference between A, and Az varies significantly, once levels of other factors change, effect of A tends to be significant. If OA technique is used, a factor having consistent effect with different conditions of other factors will be significantly estimated. That means a large factorial effect is obtained from OA experimentation ( or the order of the preferable level ) that does not vary even if there is some variation in the levels of other factors. 4.4 The orthogonal arrays being discussed in this standard are for 2" and 3" series, that is, all the factors are either at two levels or three levels. The procedure for experiments with factors at different levels is also discussed. The orthogonal array tables for 2" series and 3" series are given in Annexes A and B respectively. For 2" series, the orthogonal array tables are given for Ld( 2S), L8( 27), Llb ( 215) and L32( 231) designs. For 3" series, the orthogonal array IS 10427( Part 2 ): 2006 tables are given for Lg( 34) and LZ7( 3'3 ) designs. The tables for interactions between two coiumns are also given. 5 LINEAR GRAPHS may be replicated at least twice. 6.2 In this case, each column of-orthogonal array tables given in Annex A has one degree of freedom. Therefore, one column will be used for each main effect. Similarly, as interaction between two main effects will also have one degree of freedom, one column will be used for each interaction. 6.3 The various steps for the selection orthogonal array design are given below. of 2" 5.1 Information to be derived from an experiment is not always limited to the main effects, some times interactions are also necessary. It is not very usual to design an experiment with all two-level factors or all three-level factors. If there are four-level factors, co-existing with two-or-three-level factors, it is necessary to modi~ a two-or-three-level series OA table so as to meet the requirements. Linear graphs are useful for this purpose. The linear graphs for all the OA tables are given in Annexes C and D. 6.3.1 Under the given situation, estimate the total degrees of freedom required. The total degrees of freedom are equal to the sum of degrees of freedom for main effects and interaction effects which are required to be estimated. 6.3.2 Decide, depending upon the number of degrees of freedom as to which the orthogonal array tables will be used, namely, Ld( 23), Lg( 27), L16( 2*5) or L32(23'). 6.3.3 Depending upon the situation, that is, which of the main effects and interactions are required, draw the required linear graph. 6.3.4 Select a standard linear graph from Annex C which is closest to the required linear graph. 6.3.5 Make the required changes, if any, by deleting some lines or joining nodes by lines in the standard linear graph so that the required linear graph is obtained. Write down column numbers to various main effects and interactions and obtain the design matrix. 6.3.6 Translate layout. the design matrix into physical A linear graph associated with an orthogonal array pictorially presents the information about the interaction between some specified columns of that array. Such a graph consists of a sets of nodes and set of edges, each of which joins certain pair of nodes. A node denotes a column of the array and the edge joining the two nodes denotes another column of the array which is the interaction of the pair of columns under consideration. 5.2 5.3 For example, one of the two standard linear graphs associated for Lg( 27 ) is as follows: 1 2 6 4 7 This linear graph shows that the interaction between columns 1 and 2 comes out as column 3, the interaction between columns 1 and 4 comes out as column 5 and so on. This is in line with the interaction table given after orthogonal tables in Annex A. Column 7 is shown as independent node which is apart from the triangle. This means that this column should .be allotted to that factor where interaction with the other factor is not required. It can be noted that all the columns appear as anode or an edge in the linear graph. A 3 5 0 6.3.7 Construct a random sequence of experiments to be used while carrying out the total experiment. 6.4 Example In a telephone industry, an experiment was planned to find the infience of different &omponent dimensions, on the performance of a receiver. For this purpose, it was decided to choose the following five factors, each at two levels: S1 No. ~ ii) Factor First .Level Second Level Armature thickness (A) Al= 0.73 B,= 3.41 C,= 3.41 D,= 7.995 El =26.27 Az= 0.75 Bz= 3.46 6 ORTHOGONAL ARRAY FOR 2" SERIES Pole piece height (top)(B) Pole piece height ( bottom)(C) Magnet height ( D ) Acoustic resistance (E) 6.1 The simplest case of factorial experiment is when all the factors are at ~ levels each. In the experiment, if the degrees of freedom are fully consumed by the main effects and interactions, then the degrees of freedom for error will be zero. In order to generate the degree of freedom for the error, the experiment 3 iii) iv) v) Cz= 3.46 Dz= 8.005 Ez= 30.31 1S 10427-( Part 2 ): 2006 Besides the main effects, it is also required examine the interactions AB and BC. 6.4.1 Selection of Design to h) respective factor. The theoretical design so obtained is given in Table 2. Translate the theoretical design into actual as given in Table 3. This is a saturated design as no degrees of freedom are available for estimating error. Therefore each experiment will be conducted twice to generate 8 degrees of freedom for error. Select two random sequences of numbers from 1 to 8. For this purpose reference may be made to 1S 4905. Let the random sequences are: Replication 1 :4,1, 3,2,8,7,5,6 Replication 11:6,5,2,4,7, 1,3,8 using these sequences, the 16 experiments are conducted. The various steps fQr selection design are as follows: a) Total degrees of C(l)+ =A(l)+B(l)+ +BC(1)=7 of the required required AB(l) J] freedom D(l)+ E(l)+ b) c) Since there are 7 degrees of freedom, this experiment may be tried in L8( 27); The required linear graph is as follows: B k) A AB A SC 00 c D Table 2 Theoretical E Design ( g) ] [ Cfause6.4.l A (2) (3) I 1 2 2 1 1 2 2 D (6) (4) 1 2 2 1 1 2 2 I d) The two standard linear graphs (see Annex C ) for L8 ( 27) are as follows: 1 c B (1) (5) 1 1 1 ] 2 2 2 2 E (5) (6) I 2 1 2 2 I 2 ] AB (3) (7) 1 ] 2 2 2 2 ] 1 E (7) (8) I 2 2 ] 2 1 1 2 2 6 3f7 ment No. (-t ) I 2 3 4 5 6 7 8 (4) (2) I 2 1 2 I 2 I 2 A 3 5 -o 47 2 6 Y 5 4 4 e) In this case both the standard linear graphs can be used with equal ease. The changes required in the standard linear graphs areas follows: A.oYa 2 6 467 NOTE -- Columns (3) and (5) are used only for the computation and not in the actual conduct of the experiment. So these columns do not appear in the physical layout. Table 3 Actual Design [Clause 6.4.l (h ) ] Experiment No. (1) 1 2 0 Allocation of main effects and interactions to various columns is as follows: B:l t A (2) 0.73 0.73 0.75 0.75 0.73 0.73 0.75 0.75 (3) 3.41 3.41 3.41 3.41 3.46 3.46 3.46 3.46 Level of Factors A BC (4) 3.4.1 3.46 3.41 3.46 3.41 3,46 3.41 3.46 D (5) 7.995 8.005 8.005 7.995 7.995 8.005 8.005 7.995 E (6) \ AB:3 A:2 A: BC:5 C:4 26.27 30.31 30.31 26.27 30.31 26.27 26.27 30.31 ~6 o E:? 3 4 5 6 7 8 g) As the columns 1,2,4,6, and 7 are allotted to factors B, A, C, D and E, write down these columns from orthogonal tables for L8 (27) from Annex A, and above each column, the 4 IS 10427( Part 2 ): 2006 6.4.2 Analysis The various steps in the analysis of above designed experiment are as follows: a) Let Y,, Y2, ... ... ... .....ygbe thetest responses in the first replication andyl `,y2', ... ... ... .... y~' in the second replication for experiments 1,2, . .. .. . . .. . . .... 8 respectively. Denote ~ = (yi +X') as total response from ith experiment for both the replications. Prepare the total and.average response table for the main effects and interactions, with the help of theoretical design obtained in 6.4.1 as given in Tables 4 and 5. Sums of squares for the main effects are obtained from CO I 2 of Table 4. For example, Grand total = G = z (yi +Yi') Correction factor (CF)=G2/16 Sum of squares due to factor A, ( SSA ) = {( T2A1+T2A2)/8}-CF Table 4 Response Table for Main Effects [ Clause 6.4.2(b) Factor Level (1) A, A2 B, B2 Total Response (2) T,+ T2+T5+T6(=TA, T3+T4+T7+T8(=TA2) TI+T2+T3+T4(=TB, T5+T6+r7+T8(=TB2) TI+T3+T5+T7(=TC, T2+T4+T6+T8(=TC2) TI+T4+T5+T8(=TD1) T2+T3+T6+T7(=TD2) T1+T4+T6+T7(=TE1) T2+TJ+T5+T8(=TE, ) ) ) ) and(h)] Average Response (3) (T, +T2+T5+T6 (T, +T4+T7+T8 (T, +T2+T3+T4 (T5+T6+T7+T5 (T, +T3+T5+T7 (T2+T4+T6+T5 )/8 )/8 )/8 )/8 )/8 )18 b) c, C* D, D2 E, E2 (T, + Td+ T5+ T5)/8 ( T2+ T3+ T6+ T7)18 (T, + Td+ T6+ T7)/8 ( T2+ T3+ T5+ T5)/8 c) Table 5 Table for Interaction [ Clause 6.4.2(b) and(h)] Factor Total Response Effects Combination Level (1) Al B, A,B2 A2B, A2Bz B, C, B, C2 B2C, B2C2 (2) Tl+Tz(=TA, T5+T6(=TA; T3+T4(=TA2~, T7+T8(=TA2B2) Average Response (3) The sum of squares for other main effects may also be obtained in a similar way. d) e) Total sum of x(y2i+y2i')-cF squares ( TSS ) = ~,) ~;) ) ( T, + T2)/4 ( T5+ T6)/4 ( T3+ T, )/4 T7+ T* )/4 Sum of squares due to interaction effects AB (~~m)= { [T2A1 ~1+ T2A1~2+T2A2B,+ T2A2B2]/4 ] - CF - SSA - SS~ TI+TJ(=TB, T2+T4(=TB, T5+T7(=TB2C, C,) C2) ) T, + T3)/4 T2+ Td)/4 T5+ T7)/4 T6+ T8)/4 The sum of squares due to interaction BC may be obtained in similar way. o Sum of squared due to error ( SSE ) = TSS ­ SSA ­ SS~ ­ SSC ­ SS~ ­ SS~'­ SSA~ ­ SSBC Tb+T8(=T~2C2) Table 6 Analysis of Variance Table [Clause Source of Variation (1) Main effects 6.4.2 (f)and (g)] F Ratio 1(5)[=(4)/(3)1 MsAmsE The above sum of squares may be entered in the analysis of variance table ( Table 6 ). g) The mean squares due to main effects or interactions, as obtained in Table 6, are compared with mean square due to error and the significance of the main effects and interactions are tested. For the main effects and interactions, which are significant, the optimum level is chosen with the help of average response tables ( see Tables 4 and 5 ). The level for which the response is optimum ( maximum or minimum, as the case maybe) is selected. For other non-significant factors and interactions, the level for which the costitime/ Iabour is minimum, is selected. Sum of Degree of Mean Square Freedom Square (2) (3) (4) MSA Factor A Factor B Factor C Factor D Factor E I SSA SSB Ssc SSD SSE I 1 1 I I MS~ MSC MS~ MS~ M3BIMSE MS~ MSE MS~l MSE MS~/MSE h) I Interactions AB I I BC 1 k I Error I I I II SSAB "Ssnn .. TS; I I I I I MSAB MSm/ MSE I I I I 15 MS~C MSBC/ MSE M.V. ..- 1ss.18 1 I E I 1 Total 1 I I I 5 IS 1-0427( Part 2 ): 2006 6.5 Example For minimizing the value of the tan 5 of HV insulation system, the following four factors each at two levels, were studied: kind of conducting tapes -- adhesive or non-adhesive, thickness of conducting tapes, curing temperature, and curing pressure. From tetihnical considerations, it was felt that the interactions AC and CD may exist. For this purpose, Lg( 27) experiment was conducted and the responses ( coded) are given in Table 7. Set up an analysis of variance table, examine the significance of main effects and interactions AC and CD; and find the optimum level for each factor. 6.5.1 For obtaining the sum of squares due to main effects and interactions, the total and average responses are calculated as given in Table 8. From the response table, a) b) c) Total sum ofsquares ( TSS ) = Z Y*­ CF = 281.96 Correction factor ( CF ) = ( 1339 =74 705.04 )2 /24 =6.5.3 The tabulated value of F for ( 1, 17) degrees of freedom at 5 and 1 percent level of significance is 4.45 and 8.40 respectively [ see IS 6200 ( Part 1 ) ]. Since the calculated value of F for the main effect A k more than the tabulated value at 1 percent level of significance, this factor is highly significant. Similarly as the calculated value of F far the interaction AC is more than the tabulated value at 5 percent level of significance, this interaction is significant. 6.5.4 As the aim is to minimize the value of tan 6, the second level of factor, A which gives the lower response, is selected as optimum level. Similarly for interactions AC, which is significant, the combination A2 Cl has minimum value. Hence first level of factor C is selected. For the other two factors, namely, B and D, the cost considerations may be taken into account for selection of level. 7 ORTHOGONAL ARRAY FOR 3" SERIES g) SSAc=[( 3442+ 3092+ 3512+ 3352 )/6]­ SSA-SSC - CF=45.38 h) j) SSc~ = [ ( 3322+ 3392+ 3282 + 3402)/6 ]-SSC -SS~-CF=0.3-8 Sum of squares due to error ( SSE ) = TSS SS~-SS~ -SSc-SS~ -SSAc­SSc~ = 108.40 6.5.2 The above sums of squares maybe entered in the analysis of variance table ( see Table 9 ). Sum of squares due to factor A ( SSA ) = (6442 +6952 )/12-CF= 108.38 ]-CF d) SS~= [ (6652+6742)/12 e) 0 SSC= [(6792 +660z)/12]­ SS~=[(6722+ `3.38 1.00 CF= 15.04 6672)/12 ]-CF= 7.1 In this case, each column of orthogonal tables ( see Annex B ) has 2 degrees of freedom. Therefore for each main effect ( as it has two degrees of freedom ) one column will be used, whereas two columns will be used by an interaction, as it has four degrees of freedom. Table 7 Test Responses ( Clause 6.5) Experiment No. 2 1 Factor A (1) I Column Number \ 4 7 Response ( Hardness ) . Replicate 3 Total Replicate 1 Replicate 2 B (3) B, B2 B, Bz B, B2 B, B2 c (4) c, c, c, c, C2 C* c1 C* D (5) D, Dz Dz D, D2 D, D, Dz (6) (7) 60 60 58 49 57 58 51 55 (8) 57 '56 50 52 57 58 58 57 (9) 178 173 159 150 171 173 166 169 1339 (2) A, A, Az A2 A, A, AZ AZ 61 57 51 49 57 57 ~7 57 2 3 4 5 6 7 8 Total 6 IS 10427( Table 8 Total and Average Response ( Clause 6.5.1 ) Part 2 ): 2006 of the main effects and interactions are required, draw the required linear graph; d) Select a standard linear graph from Annex D which is closest to the required linear graph; Make the required changes, if any, in the standard linear graph so that the required linear graph is obtained. Write down column numbers to various main effects and interactions and obtain the theoretical design; Translate the theoretical design into physical layout; and Construct a random sequence of experiments to be used while carrying out the total experiment. Factor Level (1) Al A2 B, 4 c, C2 D, Dz A,C, AIC2 A2C, A,C2 C, D, C, Dz C2DI C2DZ Total Response (2) 178+173+ 159+ 150+ 178+ 159+ 173+ 150+ 178+ 173+ 171+ 173=695 Average Response (3) 695112= 57.9 644/12= 53.7 674/12= 56.2 665/12= 55.4 660/12= 55.0 679112= 56.6 667/12= 55.6 672/12= 56.0 351/6 = 58.5 344/6 = 57.3 309/6 = 51.5 335/6 = 55.8 32816 = 54.7 33216 = 55.3 339/6 = 56.5 340/6 = 56.7 e) 166+ 169=644 171 + 166=674 173+ 169=665 159+ 150=660 o g) 171 + 173 + 166+ 169=679 178+ 150+ 173+ 166=667 173+ 159+ 171 + 169=672 178+ 173=351 171 + 173=344 159+ 150=309 7.3 Example In an investigation for obtaining required colour in a watch dial at plating stage, following four factors were studied, each at three levels. Design an experiment for studying the main effect and interactions AB, AC and BC. S1 No. ~ Factor Levels 166 + 169=335 178+ 150=328 !73 +159=332 173+ 166=339 171 + 169=340 Table 9 Analysis of Variance Table ( Clause 6.5.2) Source of Variation (1) Main Effects Factor A Factor B Factor C Factor D Interactions IAC CD Error Total I 11 1 1 1 I 108.38 3.38 15.04 1.00. I 108.38 I 3.38 15.04 1.00 ] 17.00 I 0.53 I 2.36 0.16 Degrees of Freedom (2) Sum of Squares (3) Mean Square (4) F (5) 47.50c, 550c A = Temperature of bath 400c, B = Voltage ii) iii) iv) 3.5V,4.5V, 5.5V 30s,40s,50s C= Time of immersion D = Concentration of 30%,35%, 40% bath 7.3.1 Design The various steps in giving the layout of the design are as follows: a) 1 1 17 23 I 45.38 0.38 108.40 281.96 I 45.38 0.38 6.38 I 7. I I I 0.06 Total degrees of freedom = A(2) + B(2) + C(2) + D(2)+ AB(4) + AC(4) + BC(4) =20 Since there are 20 degrees of freedom, this ). The experiment can be tried in L27 ( 3`3 remaining 6 degrees of freedom will be used for error. The required linear graph is as follows: A b) 7.2 The various sEepsfor the selection of 3" orthogonal array design are given below: a) Under the given situation, estimate the total degrees offieedom required. The total degrees of freedom is equal to the sum of degrees of freedom for main effects and interactions which are required to Le estimated; Decide, depending upon the number of degrees of freedom, as to which of the orthogonal tables will be used, namely, L9(34)or Lz7(313); Depending upon the situation, that is, which 7 c) n b) A AB Ac o c D B BC d) c) The standard linear graph (1) for L27(313 ), given in Annex D, matches the above requirement completely. IS 10427( Part 2 ): 2006 e) The allocation of main effects and interactions to various columns is as follows: A:l 7.3.2 Analysis The various steps in the analysis of above designed experiment are as follows: a) Letyl, ye, . .. .. . .. . .. . Y27,be the test response for experiments 1,2 , . ....27 respectively Grand totai(G)=yl +yz+ .. . .. . . .. .. . +Y27, AB:3,4 A B:2 BC:8,11 AC:6,7 C:5 o D:g AB BC ;;; -- Error-- b) c) d) Error Correction factor (Cm = $ Total sum of squares (TS$ = Z# ­ CF, For obtaining the response totals due to the main effects, reference may be made to the table of theoretical design given in 7.3.l(fi. This tabb gives the information as to in which of the experiments, a particular factor is at which level. The response total for the factor A is obtained as follows: `*l=(y] +y~+ . .. . . . . .. . . . .. . .. ...+Y9) `*2=( y]~+y], + . . . . . . . .. . .. .. . .. ...+Y1*) TA3=(y,9+y20 + . .. . .. .. . .. . .. . . .. ..+y27) where A,, Az and As represent the three levels of factor A. o The theoretical design is as follows A B C D AC Experiment No. e) 1 2 5 9 3,4 1111 1122 1133 1212 1223 1231 1313 1322 1331 2112 2123 2131 2213 2221 2232 2311 2322 2333 3113 3121 3132 3211 3222 3233 3312 3323 3331 8,11 6,7 10,12,13 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 z 23 24 25 26 27 g) o g) Sum of squares due to the Factor A, ( SSA) T2A, + ~A2 + ~A3 = 9 - CF h) Sum of squares due to interaction AB, SSA~ `{[T*A1 ~,+ ~2A, B2 + ~Ar B3 + `A2 B1 + T2A2 ~2 + T2A2 ~3+ T2A3 ~1 + ~ A3 B2 + J] `A3B3 1/3 } - CF - SSA- SSB Similarly, the sum of squares due to other factors and other interactions may be obtained. k) Sum of squared due to error ( SSF ) = ­ TSS ­ SSA- ­ SSB ­ SSC - SSD ­ Sfm SSBC - SSAC 7.4 Example For determining the surface finish in a reaming operation ( .Cr ­ Mo alloy steel ), the following four factors were studied, each at three levels: A: B: The actual experiment maybe translated in similar way from the above theoretical design, as has been done in 6.4 for 2" series. The degrees of freedom for error= 26- 20= 6. So it is not necessary to go for second replication. Select a random sequence from experiment core drill size speed h) C : feed D : coolant For this purpose, an Lg [ 34) experiment was conducted. Since the 8 degrees of freedom available for this design are consumed by the 4 factors ( each having 2 degrees of freedom ), WO replications ofthk experiment were conducted. The test responses obtained are given in Table 10. 8 J> 1 to 27. For this purpose reference maybe made to IS 4905. The sequence of the experiments to be conducted shall be as per the random sequence obtained and not from experiment 1 to 27. IS 10427( Table 10 Test Responses on Surface Finish Part 2 ): 2006 ( cl~~~~ 7.4 ) Experiment No. Column 1 c r A 2 Factor B (3) 1 2 3 1 2 3 1 2 3 c (4) 1 2 3 2 3 I 3 1 2 D (5) 1 2 3 3 1 2 2 3 1 (6) 0.8 1.8 1.0 0.7 0.9 1.1 2.2 1.5 1.7 (7) 0.7 1.9 0.9 0.7 0.9 I .4 1.8 1.6 1.3 (8) 1.5 3.7 1.9 1.4 1.8 2.5 4.0 3.1 3.0 Number 3 4 . \ Replicate 1 Surface Finish Total Replicate 2 (1) I 2 3 4 5 6 7 8 9 (2) .1 1 I 2 2 2 3 3 3 Set-up an ANOVA table and examine the significance of these four factors: a) b) c) d) Grandtotal(@= l.5+3.7+ . .. . ....+9.0 =22.9 ( 22.9)2 18 Total sum of squares ( TSS ) = Z# 3.90 - CF = Correction Factor (C~ = `29.13 7.4.2 The tabulated values of F for ( 2,9 ) degrees of freedom at 5 and 1 percent level of significance are 4.26 and 8.02 respectively [ see IS 6200 ( Part 1 )]. Since the calculated value of F for the factors A and D is more than 8.02, they are highly significant. Factor -B is significant ( at 5 percent level of significance ). 8 FACTORS LEVEE 8.1 In most of the practical situations, all the -factors to be studied in an experiment do not have same number of levels. Usually an experiment consists of the combination of two-level factors, three-level factors or four-level factors. In such situations, the three-level or four-level factors may be incorporated in 2" orthogonal array designs by suitably modifying the 2" orthogonal tables. This arrangement generally gives orthogonal main effect plans and as such some precautions are to be taken when estimation of interaction effects are necessary, though in few cases it is possible to estimate the same. WITH UNEQUAL NUMBER OF Response totals for various main effects are given below: TA1= 7.1 TA2=5.7 T~l = 6.9 T~2= 8.6 T~3= 7.4 T~l = 6.3 T~2= 10.2 T~3=10.1 Tc, = 7.1 TC2=8.1 TC3= 7.7 e) T~3= 6.4 Sum of squares due to main effect A, ( SSA), = (7.1)2+(5.7)2+(10.1)2 -CF=l .68 6 9 Similarly the sums of squares due to other main effects may be calculated. The values of sum of squares of the other three main effects are as follows: SS~-=0.25 SSC= 0.08 SS~= 1.65 8.2 Experiments with Two-Level and Three-Level Factors Only The experiments wherein some of the factors are at two-levels and the others are at three-levels can be incorporated in 2" orthogonal array designs by-modifying 2" orthogonal tables or 3n orthogonal array designs by modifying 3n orthogonal tables. The procedure of suitably modifying the orthogonal tables is illustrated with the help of example given in 8.3. 9 g) The sum of squares due to error ( SS~ ) = TSS­ SSA - SS~ - SSC ­ SS~ = 0.24 7.4.1 The above sum of squares may be entered in the following analysis of variance table ( see Table 11). IS 10427( Part 2 ) :2006 . are as follows: IDLE Table 11 Analysis of Variance Table (Clause 7.4. 1) Source of Variation .(1) Core drill size (A) Speed (B) Feed (C) Coolant (D) Error Total Degree of Freedom (2) 2 2 2 2 9 17 Sum of Squares (3) 1.68 0.25 0.08 1.65 0.24 3.90 Mean Square (4) 0.840 0.125 0.040 0.825 0.027 F (5) 31.11 4.63 1.48 30.56 A 8.3 Example In an investigation on the establishment of thermal characteristics of certain type of heat sink, it was decided to examine the following four factors: Factors Thyristor type (A) Cooling device (B). Tier arrangement (~ Heat sink type (D) Levels A,, B], c,, D,, Az, Aj B2, B3 c1 Dz In the above required linear graph, column 1 is allotted to idle column, column 2 to factor A and column 4 to factor B. Since factor A has two degrees of freedom, column 3 ( which represents the interaction between columns 1 and 2 ) is omitted completely from the design. Similarly colum 5which represents the interaction between columns 1 and 4 is omitted, so as to create additional degree of freedom of factor B, The node representing the interaction is therefore encircled in the above linear graph. 8.3.2.3 A 3 2 1 5 B 4 C6 00D:? 8.3.2.4 The theoretical design as obtained is given below: Experiment No, IDLE A A B B C D (1) (2) (3) (4) (~ (q (~ It is required to examinethe main effects only. 8.3.1 This example is of 22x 32 type. The total number of degrees of freedom required are= A ( 2 ) + B ( 2 ) + C ( 1) + D ( 1) =6. Therefore, this experiment may either be tried in L8( 27) or in Lg( 34). 8.3.2 Idle Column Method In this method, thr~e-level factors are introduced in two-level series. Further the number of experiments for each level of these three-level main factors will not be same in this design and the acceptable optimum level will be repeated more number of times. Any one column in two-level orthogonal table is earmarked as idle column and no factor is allotted to it. If it is acceptable that factor A at level 2 is optimum and factor B at level 1 is optimum, then the following procedure is adopted. 8.3.2.1 Idle CGlumn Level 1 Compare A, with Az, BI with Bz 1 2 3 4 5 6 7 8 1111111 1 1.12222 1221122 1222211 2431112 2 4"3 222112.1 2222312 DD 2 3 2 1 As this design becomes saturated, two replications of the experiment will be necessary to generate 8 degrees of freedom for error. 8.3.2.5 The various steps in the analysis of above designed experiment are as follows: a) Let y,, yz, .. . .. yg, be the teSt responses in the first replication andy 'l,y'2, . .. . ...y '8. in the second replication for experiments 1, 2,...... ,8 respectively. Denote Ti=yi+Y'i = total response from ith experiment for both the replications. Sum of squares due to idle column =~ 16 [( T~+TG+TT+ T4) ]2 Tg)­(T1+T2+T3+ 2 A2with As, B1 with B3 b) This means that no change is required in the columns representing the three-level factors when idle column is at level 1. When the idle column is at level 2, replace the level 1 in column representing factor A by level 3 and replace the level 2 in column representing factor B by level 3. 8.3.2.2 The required linear graph and the allocation 10 c) Sum of squares due to factor A is obtained as follows: Sum of squares due to ( A, - A2 ) =: [T3+T4-TI-T212 IS 19427( Part 2 ): 2006 Sum of squares due =~[T7+Tg­T~­Tc]2 to (A2 ­ A3 ) Sum of squares due to factor A, ( SSA) = Sum of squares due to ( A, ­ A2 ) + Sum of squares dueto(A2­ A3) = ~ [TJ+TQ-T1-TZ]2+ T~­ TG]2 Where 2' denotes the level 2 of factor C repeated in place of level 3 in column 3 and is the dummy level. Similarly 1` denotes the first level of factor D repeated in place of level 3 in column 4. 8.3.3.-1 The various steps in the analysis of the above designed experiment are as follows: a) b) c) d) e) Letyl, y2, .. . .. yg be the test respons~ experiments 1,2 .. . . .. ...9 respectively. Grand total-(G) `y, +y2 + . . ..,Yg Correction factor ( CF ) = G2/9 Total sum of squares ( TSS ) = Z# - CF The response totals are as follows: TA1`YI + Y2+ Y3; TA2"y4 + Y5+ yb; TA3`y7 + yg+ y9; Tcl `yl + yb+ yg; TC2`Y2 + Y4+ Y9; TC2,`y3+ y~+ Y7; -- ~ [T,+ T~- for d) Similarly the sum of squares due to factor B ( ss~ ) = Sum of squares due to (B, ­ B2) + Sum of squares due to ( B] ­ B3 ) = ~ [T2+Td-Tl-T3]2+ T~­ T7]2 L~ [Tb+ T8- TB1`Yl + Y4+ Y7 TB2`Y2+ Y5+ Y8 `B3=y3+ yb+ y9 TD, =Y1+Y5+Y9 `D,.=Y3 + y4+ y8 TD2`y2 `Y6+Y7 e) Sum of squares due to factor C ( SSC) `~ [ T2+,T3+ T6+ T7­ TI­ T4­ T~­ T8J2 9 Sum of squares due to factor D ( SS~) `~ 1 [T2+T3+Tj+Ts­T~-T4­Tb­TT]2 g) Sum of squares due to error ( SS~) =+ 8.3.3 Where A,, A2 and A3 represents the three levels of factor A. Similarly other levels of factors may be defined. [( Y,­y '1)2+ (y2­y '2)2+ ........+ (y~-y'g)1 9 Sum of-squares due to factor A ( 55A) =*(rA, , + T*A2+ ~A3 ) ­ CF Dummy Level Method In this method, two-level factors are introduced in threeIevel series by suitably modify the three-level orthogonal tables. The level ( first or second ) of a two-level factors, which is expected to fare better, is repeated in more number of experiments as compared -to the other level. This is done by changing the level number 3 in a column, assigned to a two-level factor, by a level, which is expected to fare better, in a threeIevel orthogonal tables. Suppose in example in 8.3, C2 and D, are expected to fare better, then these are repeated in more number of experiments than C, and D2 respectively. Similarly, the sum of squares due to factor B may be obtained. g) Sum of squares due to factor C ( SSC) =-- (2 TC, -TC2-TC2')2 18 h) Sum of squares due to factor D ( SS~ ) = J) (2 T~2-T~, -TD,')2 18 Sum of squares to error (SS,) = ~ (TD, -T~l' )2+ A6 (TC2-TC2')2 )will The three-level standard orthogonal table L9( 34 be modified as follows: Experiment No. A B CD 8.4 Experiments Factors with Two-Level and Four-Level (1) 1 1 1 2 2 2 3 3 3 (2) 1 2 3 1 2 3 1 2 3 (3) 1 2 r 2 2' 1 ~ 1 2 (4) 1 2 ,, 1' 1 2 2 1' 1 11 1 2 3, 4 5 6 7 8 9 The experiments wherein some factors are at two levels and others at four levels can be incorporated in 2" orthogonal array designs by suitably modi&ing 2" orthogonal tables. Since the degrees of freedom for any four-level factor is three and each column in 2" orthogonal tables has one degree of freedom, three columns will be used for each four-level factor. In the linear graph, two nodes and the edge joining them make the representation of a fourlevel factor. The procedure is to choose any two columns in 2" orthogonal tables and for the pairs IS 10427( Part 2 ) :2006 (1, 1), (1, 2), (2, 1) and (2, 2) make the following transformation for four level factor: (1,1)+ 1,(1,2)+ 2,(2,1 )+3, and(2,2)+4 the smallest array in which this design can be accommodated isL16( 45). The required linear graph is shown below: The corresponding interaction column is also deleted so as to generate three degrees of freedom. 8.4.1 Example The following experiment consisting of 8 factors relates to the hydraulic design of impeller of a pump. Give the layout of the design to estimate all the main effects and interactions A x B, A x C and A x G. S1 No. ~ Factors Levels 0000 DEFH 8.4.1.2 The design matrix is obtahed from the standard linear graph 3 of the L,b( 4* ) array. This is shown as below: 2 6 3 7 --s 13 12 G6 Outlet angle of impeller (A) Inlet angle of impeller (B) Outlet width (C) Inlet width(D) Impeller eye dia (E) Vane thickness (~ Shape of vanes (G) Outlet tip length (~ 2 4 2 2 2 2 2 2 ii) iii) iv) v) vi) vii) viii) 10 K 1 9 8 o 4 n K 3 10 11 &l 9 7 13 C12 15:14? 0000 04 E5 F14 M15 8.4.1.1 The degrees of freedom for main effects is 10 and for itieractions is 5, making a total of 15. Hence, Experiment No, .A 1 B AxB 8.4.1.3 The resulting theoretical design is given as follows: 1 c AxC D E F G AxG H 2,8,10 1 2 1 2 3 4 3 4 1 2 1 2 3 4 3 4 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 3,9,11 1 2 1 2 1 2 1 2 2 2 2 1 2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 12 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 13 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 4 1 1 2 2. 1 1 2 2 1 1 2 2 1 1 2 2 5 1 1 2 2 I 1 2 2 2 2 1 1 2 2 1 1 14 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 6 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 7 1 1 2 2 2 2 1 1 2 -2 1 1 1 1 2 2 15 1 2 2 1 2 1 1 2 "2 1 1 2 1 2 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 9 ADDITIONAL 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 OA TABLES For level 2 experiments, one may refer to L ,2(2] I ) (see Annex E). For asymmetrical factorial with factors at 2 and 3 levels, one may refer to L,g( 21 x 37 ) ( see Annex F). 12 IS -10427( Part 2 ) :2006 ANNEX A [ Clauses 4.3,6.2 and 6.4.l(g) ] ORTHOGONAL ARRAY TABLES FOR 2" SERIES O.A. ( 4,3,2,2 ) LA( 23 ) 2 3 No. 1 1 2 3 4 1 1 2 2 1 2 1 2 2 1 2 1 2 Group 1 O.A. ( 8,7,2,2 ) L8( 27) 4 No. 1 2 3 4 5 6 7 8 I 1 2 3 5 6 7 1 1 1 1 1 2 2 1 1 2 2 2 2 1 2 1 2 1 2 1 2 1 2 .; 2 -1 1 2 2 1 2 2 1 .2 2 2 2 1 1 .2 2 1 -1 2 1 1 1 2 2 1 1 2 1 2 1 2 Group 1 Between ~o 1 3 Interaction CO1 Columns in L8( 27 ) 2 (:) (:) (:) (:) (;) () 3 2 4 5 6 5 4 7 6 6 7 4 5 2 7 6 5 4 3 2 (1) 13 IS 10427( Part 2 ): 2006 0.A. ( 16,15, 2,2) 8 L,6(21S) 9 10 11 12 13 I 2 2 14 15 1 2 2 SINO.1 2 I 3 4 5 6 7 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 =1 1 1 2 2 2 2 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 2 2 1 1 2 2 2 2 1 2 1 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 1 2 .1 2 1 2 2 1 2 1 I 2 1 2 2 1 1 2 2 1 2 1 2 1 1 2 2 1 1 2 2 2 2 1 2 1 2 1 1 1 1 2 2 2 2 1 1 2 2 1 2 1 2 1 2 2 1 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 2 10 11 12 13 14 15 16 Interaction Col 1 1 1 2 2 2 2 1 1 2 2 1 2 2 1 2 1 2 1 2 1 1 2 2 1 1 1 1 2 2 1 2 1 2 2 1 1 2 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 2 2 1 2 1 2 1 2 2 1 1 2 1 2 1 1 1 1 1 1 2 1 1 2 1 1 Between Columns 1 2 3 (2) 3 2 1 (3) 4 5 6 7 (4) 5 , 6 7 4 5 2 7 6 5 4 3 2 (7; -8 9 9 8 10 11 12 13 14 15 (1) 4 7 6 11 8 9 14 15 12 13 23 3 10 9 8 15 14 13 12 2 1 (5) 3 (6) 10 11 12 13 14 (; 11 10 13 12 15 14 (9; (lo) (11; 13 14 15 8 9 10 11 4 5 6 (12; 12 15 14 9 8 11 10 56 47 74 65 (13; 15 12 13 10 11 8 9 2 3 (14) 11 13 12 11 10 9 8 7 6 5 4 3 2 (15; 14 0. A.(32,31, SINO. 12345 6 7 1 1 I 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 89 1 1 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 1 2 1 1 2 2 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 i 1 2 2 1 1 10 1 1 1 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 11 1 1 1 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 12 1 1 1 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 13 1 1 1 2 2 2 1 1 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 14 1 1 1 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2,2) L3Z(23') 15 1 1 1 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 16 1 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 17 18 1 2 1 2 1 2 1 2 2 1 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 19 1 2 1 2 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 20 21 1 2 1 2 2 1 2 1 1 2 2 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 22 1 2 1 2 2 1 2 1 2 1 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 23 1 2 1 2 2 1 2 1 2 24 1 2 2 1 1 2 2 1 1 2 2 2 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 25 1 2 2 1 1 2 2 1 1 2 2 2 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 26 1 2 2 1 1 2 2 1 2 1 1 1 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 27 1 2 2 1 1 2 2 1 2 1 1 1 2 1 1 2 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 28 1 2 2 1 2 1 1 2 1 2 2 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 29 1 2 2 1 2 1 1 2 1 2 2 2 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 30 31 1 2 1 1 2 1 1 2 2 1 1 1 1 2 2 1 2 1 1 2 1 2 2 1 1 ~2e 12 2q 1s 1= 2Z ~ E 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 ~ 2 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 ? . 2 2 2 1 1 1 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 1 2 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 2 1 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 10 11 D 13 14 15 16 17 18 19 20 i 1 1 2 1 2 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 u 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 21 22 23 24 29 30 31 32 2 2 2 2 0 a Interaction Col 1 2 3254 167 76 1 3 4 5 6 7 89 98 1011 1110 13 14 15 10 11 8 9 14 15 u 13 11 Between No 12 13 14 Columns O. A. ( 32,31,2,2 15 16 17 18 19 20 ) 21 22 23 24 25 26 27 28 29 30 31 E H = 7 6 4 5 5 4 231213 2 3 1 101312151417161918 212023222524272629 28313: 9 141512131819171722 232021262724253031 2829P 151413121918171623 2221 ~ 27X25243130 3283 8 101120212223 15 8 9 16171819%29303124 252627: 12 14 1021 ~23D 17161918 18 B283130U 2427X 8 11 15 10 11 892223 ti21 13 181916173031282926 272425: 14 12 11 10 9823222120191817 16313029282726Z24 12345 6724 Z 2627282930 31 16171819 ~21D23 7625242726 W2831 301716191821 ~2322 3254 45262724X303128 W 1819161722232021 1 67 42726 Z243130W28 191817 i623222120 765 12 328 W 30312425X27 2021222316171819 32 W283130Z 2427 2621 ~23D 17161918 1303128 W26272425 Z 23202118191617 3130 W 282726252423 Z 212019181716 12345678 9 10 11 12 13 14 15 3254769 8 11 10 13 12 15 14 10 11 8 16745 91415U13 10 9 765411 8 15 14 13 12 U1314158 123 9 10 11 13 12 15 14 9 32 8 11 10 14151213101189 1 15 14 13 12 11 1098 1234 567 3 25 476 16 745 7 654 123 32 1 : IS 10427( Pati 2 ): 2006 ANNEX B ( ORTHOGONAL Experiment No. Clauses 4.4 and 7.1) ARRAY TABLES FOR 3"SERIES Column L9( 34) 1 2 3 4 5 6 7 8 9 1 1 1 1 2 2 2 3 3 3 2 1 2 3 3 1 2 3 2 3 1 3 1 2 L27(3'3) Column 4 1 2 3 3 1 2 2 3 1 1 2 3 1 2 3 Experiment No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 `1 1 1 1 -1 .1 1 1 1 1 2 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 1 1 1 1 2 2 2 3 3 3 3 1 1 1 4 1 1 1 2 2 2 3 3 3 3 3 3 1 1 1 2 2 3 2 2 2 3 3 3 1 1 1 5 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 1 1 2 3 2 2 3 1 2 3 6 1 2 3 1 7 1 2 3 1 2 3 1 2 3 3 1 2 3 1 2 3 1 2 2 2 1 2 3 "1 2 3 1 8 1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 3 1 3 3 2 3 1 3 1 2 9 1 10 1 2 3 2 3 1 3 1 2 3 1 2 1 2 3 2 3 1 2 2 1 3 1 2 1 2 3 2 2 2 3 3 3 2 2 2 3 3 3 1 1 2 3 3 3 1 1 1 2 3 2 3 1 11 1 2 3 3 1 2 2 3 1 1 2 3 3 1 2 2 3 2 1 3 3 3 1 2 2 3 1 12 1 2 3 3 1 2 2 3 1 2 3 1 1 2 3 3 1 2 3 1 2 2 3 1 1 2 3 13 ` 1 2 3 3 1 2 2 i 3 1 2 2 3 1 1 2 3 2 3 1' 1 2 3 3 1 2 2 3 1 19 20 21 22 23 24 25 26 27 2 2 2 2 2 2 2 2 z 3 3 3 3 3 3 3 3 3 2 3 2 3 1 3 1 2 2 3 1 2 3 1 2 3 2 3 1 3 1 2 1 2 1 2 3 1 3 1 2 1 2 3 1 2 2 3 2 3 1 2 2 2 17 IS 10427 (Pti2 ) :2006 ~W~ON TABLE ~R3" Column A 2 3456789 32265598 4 3 ;) 1 1 3 (:) 1 (~) DESIGN 3 13 11 r (;) 10 7 8 11 9 13 10 ;; 7 9]0 12 10 11 8 13 1123 (~!42 13 ;34 (; ? :; 12 6 13 8 12 9 11 10 5 11 7 n 6 13 11 lo 6 12 5 13 7 11 13 8 9 11 12 13 u 11 13 9 3 10 10 9 8 n 10 9 8 9 8 10 7567 13 6675 11 5756 12 4243 U 3324 11 2432 13 1234 9576 1423 (:0) 8 10 9 8 10 9 : (:1) 65 4 5 1 (R) 2 "7 1 12 1 (13) 18 IS 10427( Pati 2 ):2006 ANNEX C ( STANDARD L4 (23) (1) Clauses 5 and 6.3.4 j LINEAR 1 GRAPHS 3 2 FOR 2" SERIES o L8 (27) 1 2 6 317 (1) A 3 2 6 1 4 14 (2) 5 0 4 7 Y 5 4 10 6 12 15 L15(2'5) 4 `(1) (2) 8 2 (3) 2 6 (4) 6 Q 10 10 8 12 12 14 4 19 IS 10427( Pati 2 ): 2006 (5) (6) 4 8 10 10 40 ~2 3 08 12 010 15 6 2( ) ? 12 5 9 50 6~ 14 70 13 3 13 )1 09 011 15 14 L32(23') 19 21 (1) 26 22 25 (2) 22 24 18 16 4 1- 8 20 IS 10427( Pati 2 ): 2006 (3) 18 20 28 (4) 18 20 11 3 0 2 23 8 I 15 (5) 12 26 20 23 19 25 ~ 22' 16 31 31 28 3 17 24 2 B 8' 21 ( 7 -15 00 [ 10 18 11 21 14 IS 10427( Part 2 ): 2006 2 (6) 25 16 9 -22 17 24 H 8 31 7 I 15 0 14 0 23 (7) 2 20 18 12 14 00 13 4 10 28 15 01 16 (8) 1 22 IS 10427(Pati 2 ) :-2006 5 8 28 (9) 30 (lo) 80 90 100 24 26 30 28 2 16 19 20 3 6 4 12 13 5 70 29 31 27 25 17 18 21 22 f4 15 110 iv:: 23 1 26 (11) 24 16 07 17 22 20 21 11 23 IS 10427( Part 2 ): 2006 (12) 6 9 100 26 20 4 12 22 8 5 0 18 (13) 6 17 , ... 24 IS 10427( Pati 2): 2W ANNEX D [ Clauses 5, 7.2(d) and7.3.l ] STANDARD LINEAR GRAPHS FOR 3" SERIES LO (34) 1 1 L27(3'3) (1) 00 9 10 0 12 0 13 (2) 2 3,4 5 + 12,13 , :,10 ,1 25 IS 10427( Part 2 ) :2006 ANNEX E ( Clause ORTHOGONAL 9 ) FOR LIZ ( 211) TABLE 0. A.(-12,11,2,2) No. 3 1 2 3 1 1 1 1 1 1 1 1 4 1 1 5 1 1 6 1 2 7 1 2 8 1 2 9 1 2 10 1 2 11 1 2 4 5 1 1 2 2 2 1 2 2 2 2 2 1 1 1 2 1 2 2 1 2 1 2 2 1 6 7 8 9 10 1 2 2 2 2 2 2 1 2 1 1 1 2 2 2 2 2 2 1 2 1 1 2 2 1 2 1 2 1 2 1 1 2 2 1 1 2 2 2 1 2 2 1 2 1 2 1 2 2 2 1 1 1 2 2 1 2 1 2 2 1 1 1 2 1 1 2 2 1 2 1 2 1 .1 1 2 2 1 1 1 1 1 2 2 2 1 11 12 Group their interactions, it is necessary to analyse them one by one. there is interaction. NOTE -- Interaction between two columnsare to some extent mixed up with parts of other columns. To determine Therefore should not be used for experiments where ANNEX F ( Clause ORTHOGONAL 9) TABLE FOR L1~( 21 X 37) 3,2) 0. A.( 18,' 1 2 1 1 1 3 1 2 3 1 2 3 1 2 3 4 1 1 2 3 5 1 6 1 7 1 8 1 2 3 3 1 2 3 1 2 1 2 3 4 5 6 7 8 9 10 11 1 1 1 1 1 1 1 I 1 12 13 14 15 16 H 18 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2 2 3 2 3 I 1 2 3 3 I 3 3 I 2 2 3 1 1 2 3 1 2 3 1 2 3 2 3 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2 2 3 3 1 2 2 3 1 2 3 3 3 1 2 1 2 3 1 2 2 2 3 1 2 3 1 NOTE-- The Interactionbetweentwo columnsof level 3 is partly mingledwith there columns of level 1-3. The same remark as in case of note for L12 can be given here also. 26 IS 10427( Pati 2 ): 2006 ANNEX G (Forewora CO~~ECO~S~ION Statistical Methods for Qual@ and Reliabili~ Sectional Committee, MSD 3 Organization Indian Statistical Institute, New Delhi Ltd, New Delhi Representative (s) ) DR AJWIND SETH ( Chairman) PROF S. R. MOHAN ( Alternate SHRI Bharat Heavy Electrical Birla Cellulosic, Bharuch Continental S. N. JHA SHRI A. V. KRISHNAN ( Alternate ) ) SHRI VAIDYANATHAN SHRI SANJEEV SAOAVARTI ( Alternate SHRI NAVIN KAPUR SHRI VIPUL GUPTA ( Alternate ) Devices India Ltd, New Delhi Laser Defence Research & Development Organization, Science and Technology Centre, Delhi Directorate Electronics General Quality Assurance, Kanpur DR ASHOK KUMAR SHRI S. K. SRIVASTAVA LT-COL C. P. VIIAYAN( Alternate SHRI S. K. KIMOTHI SHRI R. P. SONDHI ( Alternate PROF A. N. NANKANA SHRI D. R. SEN ) Regional Test Laboratory (North), New Delhi ) h personal upacity (B -109 Malviya Naga~ New Delhi 11001 ~ In personal capacity (20/1 Krishna Enclave, New Delhi 110 029) Indian Agricultural Irrdian Association Kolkata Naga~ Safdaq"ung Statistics Research Institute, New Delhi for Productivity Quality and Reliability, DR V. K. GUPTA SHRI V. K. BHATIA ( Alfernate ) DR BISWANATHDAS DR DEBABRATARAY ( Alternate PROF S. CHAKRABORTY DR R. P. SURESH SHRI R. B. MADHEKAR SHRI NITIN GHAMAND1 ) `Indian Institute of Management, Indian Institute of Management, Maruti Udyog Limited, Gurgaon Lucknow Kozhikode Newage Electrical India Ltd, "Pune National Institution New Delhi Polyutrusions POWERGRID for Quality and Reliability (NIQR), SHRI G. W. DATEY SHRI Y. K. BHAT ( Alternate ) Private Limited, Kilpauk Corporation of India Ltd, New Delhi SHRI R. PATTABI SHRI SAI VENKAT PRASAD ( Afternate ) SHRI K. K. AGARWAL SHRI DHANANJAYCHAKRABoRTy ( Alternate DR S. ARVINDANATH SHRI A. K. BHATNAGAR( Alternate SHRI S. R. PRASAD SHRI KIRAN DESHMUKH SHRI DINESH K. SHARMA( Altirnate SHRI C. DESIGAN SHRI SHANTI SARUP SHRI A. KUMAR ( Alternate ) SHRI P. K. GAMBHIR. Scientist `F' & Head (MSD) ) Reliance Industries Limited, Surat Samtel Color Ltd, New Delhi Sons Koyo Steering Systems Ltd, Gurgaon SRF Limited, Manali Tata Motors Ltd, Jamsbedpur BIS Directorate General ) ) [ Representing Member Secretary SHRI LALJT MEHTA Scientist Director General (Ex-oJjcio) ] ` `D' (MSD), BIS 27 Bureau oflndian Standards institution established under the BIS is a harmonious statutory Bureau of Ind;an Standards marking and quality Act, 1986 to promote development of the activities of standardization, certification of goods and attending to connected matters in the country. Copyright BIS has the copyright of all its publications. No part of these publications maybe reproduced in any form without the prior permission in writing of BIS. This does not preclude the free use, in the course of implementing the standard, of necessary details, such as symbols and sizes, type or grade designations. 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